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I was tutoring a precalculus student, and the question at hand was asking to find the angle between two vectors, given the formula

$$\cos \theta = \dfrac {\mathbb{u} \cdot \mathbb{v}}{\|\mathbb{u}\|\|\mathbb{v}\|}$$

I don't remember what the specific vectors were, but the calculation ended up being $$\dfrac {6-2}{\sqrt {25} \sqrt {5}}$$

The first thing that occurred to me here is to rewrite the denominator as $5 \sqrt 5$. However, the book (which had a complete solution) rewrote this as $\dfrac {4}{\sqrt {125}}$. The student asked me why, and I had no answer. I know that sometimes "weird" manipulations are done on an expression to make it easier to compute without a calculator, or because it's conventional to leave the answer in a certain form in some applied field. However, $\sqrt {125}$ seems harder to compute by hand than $\sqrt 5$, and I don't know why it would ever be useful to leave it unsimplified in any applied field. I assured the student that $5 \sqrt 5$ is correct as well, but she was still frustrated because up until now she had been drilled to simplify everything. Is there any reason why the book might have done this?

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    $\begingroup$ Was there a part 2, or part b? Maybe the solutions left it that way because the author knew ahead of time that it might cancel with something else later. Otherwise, I think your answer is fine. I remember at some point I was taught that you should rationalize the denominator, so it should be $\frac{4\sqrt 5}{5}$, but it stopped mattering after high school. Haha. $\endgroup$ – Em. Jul 25 '16 at 5:41
  • $\begingroup$ @probablyme There weren't anymore parts to this question as far as I can remember $\endgroup$ – Ovi Jul 25 '16 at 5:45
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The person in the best position to answer your question is the author(s) of the book, so you might try writing to him/her/one-of-them. For my money, each of $${4\over\sqrt{125}},\quad{4\over5\sqrt5},\quad{4\sqrt5\over25}$$ has its advantages and disadvantages. The author had to choose one (I suppose $4(5)^{-3/2}$ is an option, too), and whichever one the author might choose, someone would ask, why choose that one? Try to reassure the student that, until and unless she has been given detailed and unambiguous instructions as to what constitutes "simpler", any one of the alternatives will be accepted and given full credit.

Me, I've never been able to figure out whether $(a+b)(a-b)$ simplifies to $a^2-b^2$, or whether $a^2-b^2$ simplifies to $(a+b)(a-b)$.

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    $\begingroup$ For me, I've always wondered whether it is better to write "$\frac12 + \frac{\sqrt{3}}{2} i$" (separate real and imaginary parts) or "$\frac{1+\sqrt{3}i}{2}$" (single division by two)... $\endgroup$ – user21820 Jul 27 '16 at 10:01

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